where \(\mathbf {x}_T,\mathbf {z}_t\sim \mathcal {N}(0,I_{C\times H\times W})\)  and \(\odot \)  denotes element-wise product.
The linear operator \(D^{-1}\)  is the inverse of two-dimensional DCT \(D\) .
Critically, the space \(M_{\text{space}}\)  and frequency \(M_{\text{freq}}\)  filters, the key design of TDAS, are target distribution aware with the same shape as \(\mathbf {x}_T\) .
Their functions here are to regulate the coordinates of initial sample and noises in their respective domain,
subject to the prior structural information of image distribution.
Note that now the covariance matrix of \(\mathbf {\eta }_t\)  can be any general positive definite matrix.
Seemingly, TDAS had a risk of changing the convergence distribution of the diffusion process, according to an analysis of its Fokker-Plank equation [1]}.
This analysis is only verified by assuming that (1) the step size is small enough and (2) the SGM learns the exact score function. However, both are invalid in our context: the step size is large under acceleration, and the SGMs often only approximate the ground-truth score function near a sub-manifold [2]}.
Hence we consider the deviation from the target distribution as analyzed above dominates and
forms a major obstacle for fast convergence (Theorem (REF )).
